{"id":244,"date":"2017-08-23T10:30:56","date_gmt":"2017-08-23T10:30:56","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=244"},"modified":"2017-08-31T04:55:49","modified_gmt":"2017-08-31T04:55:49","slug":"pythagorean-trigonometric-identities","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/pythagorean-trigonometric-identities\/","title":{"rendered":"Pythagorean Trigonometric Identities"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/introduction-to-tangent-cotangent-secant-and-cosecant\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/amplitude-frequency-period-and-phase-shift\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Pythagorean Trigonometric Identities<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, we will define and thoroughly investigate the Pythagorean trigonometric identity.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>There are six <em><strong>trigonometric\u00a0ratios <\/strong><\/em>that can help you to solve for lengths of\u00a0sides in right triangles.<\/li>\n<li><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image003.gif\" width=\"127\" height=\"44\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image006.gif\" width=\"129\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image009.gif\" width=\"112\" height=\"44\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image012.gif\" width=\"112\" height=\"44\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image015.gif\" width=\"128\" height=\"44\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image018.gif\" width=\"128\" height=\"44\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/strong><\/li>\n<li><span style=\"text-decoration: none;\">The Pythagorean\u00a0Theorem states that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p1_clip_image021.gif\" width=\"83\" height=\"21\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in right triangles (where side <\/span><em><span style=\"text-decoration: none;\">c\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the hypotenuse, and sides <\/span><em><span style=\"text-decoration: none;\">a\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">b\u00a0<\/span><\/em><span style=\"text-decoration: none;\">are legs). <\/span><\/li>\n<\/ul>\n<section>\n<h3>What is a trigonometric identity?<\/h3>\n<p><strong>Trigonometric identities<\/strong> are equations that\u00a0are always true and can simplify complex functions.<\/p>\n<p class=\"lesson_subhead\">What is a Pythagorean trigonometric identity?<\/p>\n<p>The trigonometric identity involving cosine and sine is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image003.gif\" width=\"119\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This identity is known as the Pythagorean trigonometric identity\u00a0because it can be derived using the Pythagorean Theorem, and, in\u00a0turn, the Pythagorean Theorem can be derived using the Pythagorean\u00a0trigonometric identity. There are two other identities, referred\u00a0to as Pythagorean identities, that can be derived using this\u00a0identity.<\/p>\n<p>You should be able to derive the Pythagorean trigonometric\u00a0identity using the Pythagorean Theorem.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image006.gif\" width=\"83\" height=\"21\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using the trigonometric ratios and the triangle below, we can say that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s5_p2_html_m74688601.gif\" width=\"275\" height=\"29\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20005.JPG\" alt=\" Pythagorean Theorem\" width=\"200\" height=\"108\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Divide both sides by the hypotenuse squared.<\/p>\n<p align=\"CENTER\"><em>1 =\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image009.gif\" width=\"213\" height=\"52\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/><\/em><\/p>\n<p>Since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image012.gif\" width=\"127\" height=\"44\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image015.gif\" width=\"129\" height=\"44\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we can use substitution and get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image017.gif\" width=\"119\" height=\"21\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>If you begin by assuming that the Pythagorean trigonometric\u00a0identity is true, can you work backwards and end up with the\u00a0Pythagorean Theorem?<\/p>\n<p>Now, we will hit the brakes and try to do what we did above,\u00a0but in reverse. We will try to end up with the Pythagorean\u00a0Theorem, beginning with the identity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image019.gif\" width=\"119\" height=\"21\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This process may be a bit too complicated to attempt on your own,\u00a0so we will go through this one together.<\/p>\n<p>Start by using the identity<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image021.gif\" width=\"119\" height=\"21\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>On the unit circle below the\u00a0<em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">&#8211;\u00a0and <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axes,\u00a0we <\/span>can see\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image024.gif\" width=\"37\" height=\"19\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image027.gif\" width=\"35\" height=\"19\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0on the right triangle inside of the unit circle. The easiest way\u00a0to continue is to pull the right triangle out of the circle for\u00a0evaluation purposes.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/Math%20Mod%206.2%20Art%20006.JPG\" alt=\"Unit circle with triangle removed\" width=\"337\" height=\"160\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>If we use substitution, we get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image030.gif\" width=\"75\" height=\"21\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>And using the basic trigonometric ratios, we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image033.gif\" width=\"105\" height=\"44\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image036.gif\" width=\"104\" height=\"44\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>So we can use substitution again to get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image039.gif\" width=\"237\" height=\"52\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image042.gif\" width=\"208\" height=\"48\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Now multiply both sides by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image045.gif\" width=\"83\" height=\"24\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image048.gif\" width=\"241\" height=\"24\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">And since the side opposite\u00a0of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image051.gif\" width=\"13\" height=\"19\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is <\/span><em><span style=\"text-decoration: none;\">a<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0the side adjacent to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image053.gif\" width=\"13\" height=\"19\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is <\/span><em><span style=\"text-decoration: none;\">b<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0and the hypotenuse is <\/span><em><span style=\"text-decoration: none;\">c<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0we get <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image056.gif\" width=\"83\" height=\"21\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">And si<\/span>nce\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image058.gif\" width=\"83\" height=\"21\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the Pythagorean Theorem, we have derived the Pythagorean\u00a0Theorem from the trigonometric identity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p2_clip_image060.gif\" width=\"119\" height=\"21\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<h4>What are the other two Pythagorean trigonometric identities\u00a0that can be proved using the Pythagorean trigonometric identity\u00a0that involves sine and cosine?<\/h4>\n<p>The other two trigonometric identities that can proved using\u00a0the Pythagorean trigonometric identity that involves sine and\u00a0cosine are <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image003.gif\" width=\"121\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image006.gif\" width=\"120\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Try and prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image008.gif\" width=\"121\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0using the Pythagorean trigonometric identity that involves sine<br \/>\nand cosine.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image011.gif\" width=\"120\" height=\"21\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Now divide both sides by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image014.gif\" width=\"45\" height=\"21\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and you get:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image017.gif\" width=\"128\" height=\"44\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image020.gif\" width=\"145\" height=\"49\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>.<\/p>\n<p>I<span style=\"text-decoration: none;\">n the previous section,\u00a0you learned that <\/span><strong><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image023.gif\" width=\"88\" height=\"41\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">and that\u00a0<\/span><strong><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image026.gif\" width=\"88\" height=\"41\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">So, using\u00a0substitution, you find that:<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image029.gif\" width=\"137\" height=\"25\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image031.gif\" width=\"121\" height=\"21\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Try to derive the trigonometric identity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image033.gif\" width=\"120\" height=\"21\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0beginning with the Pythagorean trigonometric identity that\u00a0involves sine and cosine.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image035.gif\" width=\"120\" height=\"21\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Divide both sides by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image038.gif\" width=\"43\" height=\"21\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image041.gif\" width=\"161\" height=\"44\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image044.gif\" width=\"153\" height=\"49\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Since we learned in the\u00a0previous section that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image047.gif\" width=\"88\" height=\"41\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and that <\/span><strong><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image050.gif\" width=\"85\" height=\"41\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">we can use\u00a0substitution to get: <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image052.gif\" width=\"120\" height=\"21\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>These three trigonometric identities need to be memorized. But if you\u00a0can derive two of the three from the Pythagorean trigonometric identity involving sine and cosine, then\u00a0there are two less formulas you will need to remember. This is why it is a good idea to know how to\u00a0derive formulas and identities. If you forget an identity, you can always derive it from another one you\u00a0know!<\/p>\n<\/div>\n<p>These are the three most well known identities, but there are\u00a0many more, some of which will be discussed in a later section.<\/p>\n<p class=\"lesson_subhead\">How do you use these trigonometric identities?<\/p>\n<p>As we discussed earlier, a trigonometric identity is an\u00a0equation that is always true and can simplify a complex equation.\u00a0The best way of learning how to use these identities is to\u00a0practice, practice, practice.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the correct simplification of the expression\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image055.gif\" width=\"157\" height=\"23\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image058.gif\" width=\"37\" height=\"19\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image061.gif\" width=\"48\" height=\"19\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>\u20131<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p style=\"text-decoration: none;\">The correct choice is C.<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image064.gif\" width=\"155\" height=\"23\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Since we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image066.gif\" width=\"88\" height=\"41\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and \u00a0<\/span><strong><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image068.gif\" width=\"85\" height=\"41\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">we can use\u00a0substitution.<\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s5_p3_html_m24141284.gif\" width=\"169\" height=\"184\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p style=\"text-decoration: none;\">Using the Pythagorean\u00a0trigonometric identity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image074.gif\" width=\"121\" height=\"21\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image077.gif\" width=\"132\" height=\"21\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0If we use substitution, we find that<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p3_clip_image083.gif\" width=\"70\" height=\"100\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<p>You can look at using these\u00a0trigonometric identities as a game. The better you are at using\u00a0these, the easier trigonometry is going to be. So try making up\u00a0your own expressions, and try simplifying them using Pythagorean\u00a0trigonometric identities.<\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li><em><strong>Trigonometric\u00a0identities<\/strong><\/em> are equations that are always true and\u00a0can simplify complex functions.<\/li>\n<li>The <strong><em>Pythagorean trigonometric identities\u00a0<\/em><\/strong>are:<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p4_clip_image003.gif\" width=\"119\" height=\"21\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p4_clip_image006.gif\" width=\"121\" height=\"21\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images2\/s5_p4_clip_image009.gif\" width=\"120\" height=\"21\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/introduction-to-tangent-cotangent-secant-and-cosecant\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/amplitude-frequency-period-and-phase-shift\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<div><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Pythagorean Trigonometric Identities Objective In this lesson, we will define and thoroughly investigate the Pythagorean trigonometric identity. Previously Covered: There are six trigonometric\u00a0ratios that can help you to solve for lengths of\u00a0sides in right triangles. The Pythagorean\u00a0Theorem states that\u00a0\u00a0in right triangles (where side c\u00a0is the hypotenuse, and sides a\u00a0and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-244","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/244","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=244"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/244\/revisions"}],"predecessor-version":[{"id":597,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/244\/revisions\/597"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}